so lets say we give you a graph and tell you to do the following things

1:24

whole bunch of stuff
The limit of x approaches 4 from the minus side of f of x is 2

1:32

the limit as x approaches 4 from the plus side is 3
and f of 4 is 3
the limit when x approaches 5 from the minus side
is positive infinity
the limit as x approaches 5 from the plus side
is minus infinity
f of 0 is 0 f of negative 1 is 1
the limit as x approaches -1 is -1
the limit as x approaches infinity is 0 and the limit a x approaches -infinity is -infinity. so draw me a picture

2:01

there are many many right answers but they all have to be from the same basic ffamily
alright is that long enough i think thats long enough
so lets do this graph
so whats a good strategy
well first of all
we give you some actual points
we tell you that when f of 4 is 3 so when x is 4 y=3

2:38

we know the graph goes to 4,3
we also know its goes 0,0 also known as the origin
and it goes through negative 1,1
so that takes care, ill cross them off
so i know ive taken care of something
and now i start to look at the limits

3:02

so lets take them in order so when x approaches 4 from the minus side
f of x is equal to 2
so from the minus side
i have to get to 2 somehow
i dont know if i go up or go down it doesnt matter but at 2
i have a circle cause its not equal to 2 there its equal to 3 there
but it will be very close to 2 when x gets close to 4
now on the other side

3:31

so cross that one off
on the other side when x approaches 4 from the positive side
i go to 3, oh thats where the dot is
im doing something like that
so far so good
and you go up and down it doesnt matter at the moment
now other stuff when i approach 5 from he minus side i get infinity that means im an asymptote
i go up to positive infinity
something like that

4:02

you can squiggle on the way up it doesnt really matter
no reason to get fancy just know the curve goes up like this
because as we approach 5 from the left side
you go up positive infinity when we approach 5 from the minus side
from the positive side youre minus infinity now were down here
so there we go we took care of the whole left side
now that were there since were going to 5 now beyond 5 as x goes to infinity you go to 0

4:36

so something like that
you can go above and come down
you can jiggle and come down but like i said theres no reason to show off
no reason to screw it up just
just like that is fine
now youve taken care of almost everything

5:00

and notice f of -1 is positive 1
and as we approach negative 1 the limit is -1
so we have to have a circle then and we go through the origin
something like that
doesnt have to be perfect but some variation on that. again
we know f at 0=0
and negative 1 it equals to positive 1
but when we get very close to negative 1 from either sode

5:30

we have to get to negative 1 cause theres a whole there
because thats just the limit thats not what exactly happens at negative 1
and the last thing is
when x approaches negative infinity
y goes to negative infinity
down like that
howd we do
im gonna do another one of these
do we loce this one
yes love yes question

6:03

how do we know it goes from here rather than here because
when x gets very close to negative 1
youre at negative 1
youre only at x is positive 1 when x is exactly negative 1
then you have to get aback here again
thats how we know it comes from the whole and not the point
okay that makes sense howd you do?
love this
lets make up another 1

6:30

ill write the rules over here so that can stay there for a while

8:12

alright that was fun
you know f of 0 is 3, f of 1 is 4
the limit as x approaches 1 from the minus side is -5
the limit as you approach 1 from the plus side is 2
the limit as x goes to infinity is infinity
the limit as x approaches
negative 1 from the plus side is infinity

8:31

when it approaches -1 from the minus side it is infinity
when it approaches -3 you get -5
f of -3 is -6
and the limit as x approaches -infinity is 0
alright first thing you do is put on all the points

9:00

so f of 0
is 3
f of 1 is 4
and f of -3 is -6
we know we go through there alright so we can cross these off
lets deal with what happens when we approach 1

9:30

well when you approach 1 from the plus side
were at 2
there
and when we approach 1 from the minus side we are at 5
something like that
so dont know much else yet
so that takes care of these 2
anything going on to the right of 1
no it just goes to infinity

10:01

something like that
you can also go straight dont have the curve
you could have a
concave down curve dosent matter
how do i come to which conclusion
well the limit as x approaches infinity of f of x is infinity
as you get and x distance that will be big y distance
will be big
f of x is y because this is the graph of f of x

10:34

now something is going on at negative 1
well at netaive 1
we have to go from the plus to infinity so we have an asymptote
i dont know we can do something like that
you can make it turn in there
and when x approaches -1 from the minus side we also are infinity

11:05

so that takes care these two, i already did that
well i have to go through this point
and remember as i get very close to 1
at point 99999999 i have to be at 5
and .011111 i have to be at 2
so thats why i have those circles there
but the graph goes to those points

11:31

it just doesnt actually touch those points
but it get infinitily close
now negative 1 from the minus side i did negative infinity
and negative 3 we go to negative 5
something like that
and from the other side were at negative 5
and finally we have to go to negative infinity
as x goes to negative infity we go to 0

12:03

so some variation on that
you guys see that
something like that

12:35

we go the idea
how do we know we go through the circle at negative 5
because you know that the curve
the limit as x approaches -3
is minus 5
menas as you get closer and closer to 3 from te minus value

13:01

you have to get closer and closer -5 from the y value
so as you get closer to negative 3
you have to approach negative 5 but at -3 you leave the curve
go to negative 6 and go right back again
thats why its not continuous there
well this is 0,3

13:31

how do i know not to go through 1,4 well at 1,4 i now
its equal to 4but when i get to 1 its not equal to 4
very close to 4 its either 5 or 2 depending what side you are on
but at 1 we equal to 4
see theres no limits going on

14:00

alright im going to do one more of these
because i think you could use a third

15:44

f of is 0 f of 1 is 2 f of 3 is 6
the limit as x approaches 3 is 4
the limit as x approaches 5 from the minus side of f of x is positive infinity
the limit as x approaches 5 from the plus side is positive infinity

16:00

the limit as x approaches positive infinity is 2
the limit as x approaches 0 from the positive side is 0
the limit as x approaches 0 from the minus side is -1
and last the limit as x approaches -infinity of f of x is negative infinity
alright lets do this 1

16:37

alright what do we know about this graph
grading these are going to be entertaining, maybe we wont
maybe im just torturing you guys
alright lets put on the points that we know
we know 3 points
we know it goes through the origin because f of 0 is 0

17:04

we know f of 1 is 2
so 1,2
and f of 3 is 6
you know the curve is at those points
you get rid of that one you get rid of that one okay

17:31

as x approaches 3 were at 4
so 4 i will put that halfway between 2 and 6
so when you get close to 3 from either side
you get close to 4, by the way it can go the other way
we havent narrowed down the graph yet
do we have any other information in that area no
now what happens if we approach 5
from either side of 5 we go up

18:01

to positive infinity
so we have avertical asymptote and were gonna go up
to positive infinity from both sides
and as x approahes infinity
i have to go to 2
something like that you dont have to draw the dotted line
but it is flattening out at 2

18:31

so lets convert this to color
what happens between 1 and 3 i have no other information happening between 1 and 3
i can just connect those
now as i approach 0
from the positive side
i get to 0

19:02

and im actually equal to 0 at the origin
but when i leave 0 and im on the negative side of 0
i have to be down here at negative 1
and as x approaches infinity i go to infinity
thats all your things
we like that one were getting the hang of it? good

19:30

lets do a different type of graph

20:00

you have a spherical shape bowl
and youre going to pour water into it
this is kind of the homework question so water is going in the bowl
start filling it up
draw a graph
of the hieght of water in the bowl verses time
height verses time
so you pour water into the bowl at a constant rate
the bowl is shaped like a sphere

20:30

what is the height of the water going to look like

21:44

your graphing the height of the water verses time
if you pour in water at first its empty so its going to go up at a certain rate
then its going to change

22:09

okay so what happens, you pour wtaer in so the beginning
this is very shallow so the height goes up very quickly
as the bowl gets wider
everytime you have a certain amount of water the height goes up lesss
because it has to go out so it doesnt go ut very much

22:32

so it starts to flatten out
till you get to the middle of the bowl
then as you
get closer and closer to the top
the curve gets stepper again
until you are full
and thats whatever time you are full
okay how did we do on that
thats not so hard right

23:00

what if the bowl was shaped like a cylinder
if the bowl was shaped like a cylinder
it will get me a straight line
because at anytime it goes up the same amount
what if it was shaped like a cone

23:34

do you guys understand why when it gets wider the height and then curve flatten
because in the beginning you add lets say a gallon of water
and you fill it up this high
then your next gallon of water will fill it up this high
and your next gallon of water may only fill it up this high
you kind of flatten like that
so far so good

24:04

trying to think of some other shapes
i dont know thats enough shapes you got the idea
lets do some more graph stuff you guys love graphs

24:51

this is the graph of f verses the derivative
draw the graph as f verses x

25:00

and te graph of x verses the second derivative
so i want two graphs
first you draw the graph
x verses f
then you draw the graph x verses the second derivative so youre going to go both directions
so we have the derivative graph
so what are the keys to the derivative graph
you just have to pay attention to the things that are positive
where things are negative

25:31

when you look at a derivative graph
you dont really want to focus on the slope
you want to focus on positivity and negativity
so below the x axis means negative
above the x axis menas positive just like the other raphs
so we are below the x axis here so our slopes are negative numbers
were above the x axis here so are slope are positive numbers
so remmeber negative number means youre going down
positive numbers are going up so were going down at this spot

26:04

the derivative is 0
so somewhere around here
the derivative is 0, the graoh is going down till it gets there
and then it gets to 0
and then it turns around and goes up by the grap could be down here we dont know
and we dont care
at the origin
the slopes are all positive till we get to the origin
then it switches to negative and heres it 0 so thats must be a maximum here on our grap[h

26:35

so when you get to the y axis
we have to do something like that
okay cause the graph is going up
0 and then its going down
now all the rest fo these are negative numbers
the graph is going to go down for a while
its going to keep going down
but its gonna kind of flatten out because it has to get to 0
something like that

27:03

now so bad so thats the grpah of x
f of x given the derivative graph
now we go the other way
we want the second derivative so the second derivative
this is the original and if youre graphing the derivative
the second derivative is just the derivative fo the first derivative
so now we dont pay attention to positive and negatives we pay attention to slope
so we look at this graph and say the slope is positive

27:32

of all that side till right here
so somewhere around here
the graph of the derivative is 0 because its flat right here
so we have positive numbers from here till we get to here
here the slope would be again 0
cause negative numbers all the way to there
then we have 0 again

28:04

and thats oh there
it bottoms out right about there thats called point of inflection
then the graph goes up again positive
but it sort of turns around and gets closer to 0
so first it goes up kind of positive fast
and then it slows down till it gets to 0 again
so
that

28:34

all these slopes are positive
so they hard to start at big positive numbers and come down to there okay
alright lets do a different type of thing
this is sort of involved with graphing

29:39

okay so we have the function f of x
is the absolute value of x minus 2
divided by x minus 2
find the limit from the left as you approach 2
the limit from the right as you approach 2 and then graph the function
so whats the absolute value of 10
10 whats the absolute value of -10

30:00

10
so when you take a positive number and the absolute value of it you dont have to do anything to the number
when we have a negative number and we take the absolute value of it we make it positive
so how do you take a negative number and make it positive
you multiply it by negative 1
so if you want to take negative 10 and turn it into a positive number
multiply it by negative 1 and now it becomes positive 10
so absolute value function
you can think of as two ways

30:30

as long as x is a positive number
you do nothing
cause the absolute value of any positive number is just the number
what if its 0 well the absolute value of 0 is 0
what about the absolute value of a negative number well
we want to turn it positive so we multiply it by negative 1
another way to think of the absolute value function
Is to think of it as a peace wise function

31:02

as long as x is greater than or equal to 0
its just the function
if its less than 0 its the negative of the function
now lets think of f of x the absolute value of x-/x-2
we could say
well when is x-2 positive
when x is bigger than
2
so as long as x
as long as x is greater than or equal to 2
we dont have to do anything

31:33

so far so good
what about when x is less than 2
well when x is less than 2
this is a negative number in the absolute value bar
so just multiply it
by a negative sign
sorry thats the absolute value of that part not the hole function
so we could rewrite this

32:02

as follows
you can take the absolute value of x minus/x-2
and say that it is x minus 2 over x-2
x is greater than 2
at 2 things get interesting
and its negative of x minus 2
over x-2
when x is less than 2
but wait what is x-2/x-2

32:30

just 1 as long as x is not 2
so we can say this is 1 as long as x is greater than 2
and negative 1
as long as x is less than 2
so what is the limit as x approaches 2 from the minus side
-
1
what is the limit as x approaches 2 from the plus side
positive 1
and what is the graph of this look like

33:05

very simple graph
something like tha
thats it and
at 2 the limit does not exist and theres no value fo the function
cause you plug in 2 you get 0/0
howd you guys do on that one

33:45

youre asking is it a bad sign to coming from the right side to make x 3
well to plug in 3
you get 1/1
you plug in 2.1 you get 1/1
or .1/.1 if you plug in .01
so you can certainly tst numbers

34:01

and you should pretty quickly come up with the idea of getting
1
everytime you plug in a number bigger than 2
and -1 everytime you plug in a number less than 1
but if you actually want to think about the algebra, thats the algebra of whats going on
alright lets do some conjinuity when we still have time

34:46

find the value of k
that makes f of x continuos if
f of x is x squared minus kx+1
when x is greater than 2

35:00

and 3kx+4 when x is less than or equal to 2
so find k that makes the function continuos everywhere
so rememebr in order to be continuous
the pencil can not leave the paper when you are graphing
that means youre graphing this equation and
you come along and get to x's
and you get closer and closer to 2 and you get some number
and then when you switch to the other piece
you have to stay
on the graph so another words when you approach 2 from the left side

35:31

you have to get to the same spot as you approached by the left side
so that means the limit as x approaches 2 from the left side
as to equal the limit as x approaches 2 from the right side
so the limit
when x approaches 2 from the minus side of f of x
equals 3k 2 plus 4
which is 6k+4

36:03

and the limit when x approaches 2 from the plus side
of f of x
well now you go to the other graph
and you plug in 2 and get 2 squared
mins k times 2 plus 1
which is 5 minus 2k
so five minus 2k
has to equal 6k plus 4
thats a five

36:32

so 6k plus 4
has to equal 5 minus
2k
so 8k
has to equal 1
k has to equal 1/8
lets do one other
its just 1/8 k equals 1/8 thats all you have to do you solved the problem

37:11

all we are asking for is the value of k that makes it continuous
1 last quick one

37:50

what if we asked where is this function continuous
so we have x squared -3x+1 as long as x is less than or equal to 1

38:03

x+1 when x is between 1 and 2
and 4x minus 5
when x is greater than or equal to 2
where is the function continuous generally the function is continuos everywhere
if theyre polynomial
your only problem are going to occur at 1 and at 2
so at x equals 1
from the left side

38:31

the limit as x approaches 1 from the left side
is 1 minus 3 plus 1
is negative 1
the limit
when x approaches 1 from the plus side
is 2 so its not continuous at x=1
i plugged 1 into the top branch
and i plugged one into the middle branch
and i dont get the same value so its not continuous at x equals 1

39:03

now what happens at x equals 2
well lets do the limit as x approaches 2 from the minus side
of f of x
and i get 3
and i do the limit
when x approaches 2 from the plus side of f of x
and i get 8-5 is 3
so it is continuous at x=2
so if i said wheres this function continuous i would say all reals

39:36

except
x equals 1
you can write that in fancier notation if you want to
alright study ahrd